Second Derivative Calculator

Calculate the second derivative f''(x) numerically using a 5-point stencil. Tests concavity (concave up/down), finds inflection point candidates, and computes curvature κ and Taylor coefficients.

f''(x) at x (5-point stencil)
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f'(x) —
f(x) —
Concavity —
Extended More scenarios, charts & detailed breakdown ▾
f''(x)
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f'(x) —
Concavity —
Professional Full parameters & maximum detail ▾

Higher Derivatives

f'''(x) third derivative —
f''''(x) fourth derivative —

Taylor & Curvature

Taylor coefficients a₀..a₄ —
Curvature κ = |f''|/(1+f'²)^1.5 —

Critical Point Test

Second derivative test —

How to Use This Calculator

  1. Enter the function f(x) (e.g. x^3−3*x).
  2. Enter the point x.
  3. Get f''(x) via 5-point stencil, f'(x), f(x), and concavity.
  4. Use the Concavity Test tab to test multiple points in an interval.
  5. Use the Inflection Points tab to scan an interval for sign changes in f''.

Formula

f''(x) ≈ [−f(x+2h)+16f(x+h)−30f(x)+16f(x−h)−f(x−2h)]/(12h²)

Curvature κ = |f''|/(1+f'²)^(3/2)

Example

f(x)=x³−3x at x=0: f''(0)=6·0=0 (inflection at x=0); at x=1: f''(1)=6·1=6>0 (concave up).

Frequently Asked Questions

  • The second derivative f''(x) is the derivative of the first derivative f'(x). It measures how quickly the slope is changing — the rate of change of the rate of change.
  • If f''(x) > 0 at a point, the function is concave up (cup shape) there. If f''(x) < 0, it is concave down (cap shape). If f''(x) = 0, it may be an inflection point.
  • At a critical point where f'(x₀)=0: if f''(x₀)>0 it is a local minimum; if f''(x₀)<0 it is a local maximum; if f''(x₀)=0 the test is inconclusive.
  • An inflection point is where the concavity changes (from up to down or vice versa). At such a point f''(x)=0 (necessary but not sufficient — you must verify a sign change in f'').
  • Curvature κ = |f''|/(1+f'²)^(3/2) measures how sharply a curve bends. A straight line has κ=0; a circle of radius r has κ=1/r everywhere.

Related Calculators

f' Derivative Calculator d/dx Implicit Diff. — Tangent Line ∑ Taylor Series
Sources & References (5) ▾
  1. Second Derivative — Paul's Online Math Notes — Lamar University
  2. MIT OCW 18.01 — Concavity — MIT OpenCourseWare
  3. Stewart's Calculus — Concavity and Inflection — Cengage / Stewart
  4. NIST DLMF — Numerical Differentiation — NIST
  5. OpenStax Calculus Vol. 1 — Ch. 4 — OpenStax